Abstract: This paper is devoted to the study of rigidity properties for specialsolutions of nonlinear elliptic partial differential equations on smooth,boundaryless Riemannian manifolds. As far as stable solutions are concerned, wederive a new weighted Poincar\-e inequality which allows to prove Liouvilletype results and the flatness of the level sets of the solution in dimension 2,under suitable geometric assumptions on the ambient manifold.